Question: Arc $AC$ is a quarter-circle with center $B$. The shaded region $ABC$ is "rolled" along a straight board $PQ$ until it reaches its original orientation for the first time with point $B$ landing at point $B^{\prime}$. If $BC = \frac{2}{\pi}$ cm, what is the length of the path that point $B$ travels? Express your answer in simplest form.

[asy]

filldraw((0,0)--(-1,0)..dir(135)..(0,1)--(0,0)--cycle,gray,linewidth(2));
draw((0,1)..dir(45)..(1,0),dashed);

draw((1-7/25,24/25)--(1+17/25,31/25)..(1-7/25,24/25)+dir(-30)..(1,0)--(1-7/25,24/25)--cycle,dashed);

draw((3.5,0)--(2.5,0)..(3.5,0)+dir(135)..(3.5,1)--(3.5,0)--cycle,dashed);

draw((-1.5,0)--(4,0),linewidth(2));

label("P",(-1.5,0),W);
label("A",(-1,0),S);
label("B",(0,0),S);
label("C",(0,1),N);

label("A$^{\prime}$",(2.5,0),S);
label("B$^{\prime}$",(3.5,0),S);
label("Q",(4,0),E);
[/asy]
Solution: We can divide the rolling into four phases:

Phase 1: The quarter circle pivots $90^\circ$ about point $B$. [asy]
pair A = (-1,0); pair B = (0,0); pair C = (0,1);
path q = B--A..dir(135)..C--cycle;
draw( (-1.5, 0)--(1.5, 0), linewidth(2) );
filldraw( q, gray, linewidth(2) );
draw(rotate(-90)*q, dashed);
label("$A$", A, S); label("$B$", B, S); label("$C$", C, N);
[/asy] In this phase, point $B$ does not move.

Phase 2: The quarter circle pivots $90^\circ$ about point $C$. [asy]
pair A = (0,1); pair B = (0,0); pair C = (1,0);
path q = B--A..dir(45)..C--cycle;
draw( (-0.5, 0)--(2.5, 0), linewidth(2) );
filldraw( q, gray, linewidth(2) );
draw(rotate(-90, (1,0))*q, dashed);
label("$A$", A, N); label("$B$", B, S); label("$C$", C, S);
[/asy] In this phase, point $B$ is always $\frac{2}{\pi}$ cm away from point $C$, so its path is a quarter-circle with radius $\frac{2}{\pi}$. The circumference of a circle with radius $\frac{2}{\pi}$ is $2\pi(\frac{2}{\pi}) = 4$, so $B$ travels $\frac{1}{4}(4) = 1$ cm.

Phase 3: The quarter circle rolls along arc $CA$. [asy]
pair A = (1,0); pair B = (0,0); pair C = (0,-1);
path q = B--A..dir(-45)..C--cycle;
draw( (-0.5, -1)--(2.07, -1), linewidth(2) );
filldraw( q, gray, linewidth(2) );
draw(shift((1.57,0))*rotate(-90)*q, dashed);
label("$A$", A, N); label("$B$", B, N); label("$C$", C, S);
[/asy] In this phase, $B$ is always $\frac{2}{\pi}$ away from the ground, so its path is a straight line segment parallel to the ground. From the diagram, we see the length of this line segment is equal to the distance between the original position of $C$ and the new position of $A$. This distance is traced out by arc $CA$ as it rolls. So its length is the length of arc $CA$, which is 1 cm (since it's a quarter of a circle with radius $\frac{2}{\pi}$, a length we've already calculated). So the path of $B$ has length 1 cm.

Phase 4: The quarter circle pivots $90^\circ$ about point $A$. [asy]
pair A = (0,-1); pair B = (0,0); pair C = (-1,0);
path q = B--A..dir(-135)..C--cycle;
draw( (-1.5, -1)--(1.5, -1), linewidth(2) );
filldraw( q, gray, linewidth(2) );
draw(rotate(-90, (0,-1))*q, dashed);
label("$A$", A, S); label("$B$", B, N); label("$C$", C, N);
[/asy] As in phase 2, the path of $B$ has length 1 cm.

Putting this together, the path of point $B$ has total length $1 + 1 + 1 = \boxed{3\text{ cm}}$.